1. I think you misunderstood my point. So, have you tried out what I suggested and did that work for you? I think it should, otherwise it may mean you need to explicitly specify the classpath in the java command, which is tedious and I will avoid it if ever possible because, depending on the Java project, missing some Java library class paths that exist in the default classpath will render those libraries unloadable.

No, the Java classpath has not changed. Unless you explicitly specify the classpath, the default classpath used by java ALWAYS contains the current directory (.) that's why if you put a Java class that is not in any package (see Java books for "package" if you don't know what Java packages (i.e. namespaces) are and how they work) in the current directory Java can always find it because . is in the classpath. But when you run it from cron, the current directory is not where your class resides, so it cannot be found. That was the point I would like to make in my previous reply.

PATH has nothing to do with class invocation in java UNLESS you install java in non-standard locations. You can qualify java with absolute path but as you said previously it did not work either. Therefore, it seems like Java can be loaded, it just couldn't find the class. Java doesn't use PATH to do anything internally except from to load the java executable itself.

2.
To same file:
java MyClass >~/myLogfile 2>&1

To different file:
java 1>~/stdout.log 2>~/stderr.log

But if for use with cron, I recommend avoid using ~ but to use absolute paths for filenames.